Determine the resulting angle of a reflected sprite (diagonal mirror)

Question

I am trying to achieve a simple reflected sprite effect. Imagine a diagonal line which goes from one corner of the screen to the other. Then, a sprite (or image) is rotated by a particular amount and placed at some location on one side of the diagonal. What would be the mathematical formula to programmatically rotate another instance of the sprite (or image) that will be placed on the reflected side of the screen? It's easy for me to figure out if the reflection is vertical or horizontal (a simple flip of the image) and I can figure out where to place the resulting sprite, but it seems like a whole different level of complexity trying to determine the resultant sprite's angle.

Any suggestions or programming formulas? I generally stink at trigonometry and can't find any leads. Again, the angle of the line of reflection is known (or can be found) as can the angle of the original sprite. I just want to determine how the reflected sprite will be rotated when displayed.

Answer 1

If I understand you properly, then it should be as simple as making sure that the reflected sprite's angle (with respect to the diagonal line) is the same as the original sprite's angle (again w.r.t. the diagonal line) but negative.

Showing this in an image (with the original sprite on the bottom left and the reflection on the top right):

So if you look at the angles θ_I1 and θ_R1, you can see that:
θ_I1 is the angle between the original image and the diagonal line
θ_R1 is the angle between the reflected image and the diagonal line

These angles are equal and opposite, i.e. 55° and −55°.

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That might not be quite enough for you to work out the rotations needed in the general case though, so I'll go into a bit more detail. Basically, like I said before, the angle between the reflected sprite and the line needs to be equal and opposite to the angle between the original sprite and the line. This is the same as making sure the difference in angle between each sprite and the line, with respect to a common reference direction, is equal and opposite.

Looking again at the image, assume that the common reference direction is to the right. It doesn't matter if you make this something else, as long as you're consistent. Also assume that the green arrows, on each sprite, would point parallel to the reference direction when the sprite has a rotation of 0°.

So with that in mind you can see that the angle of the line θ_L is 35° (with positive rotation being clockwise).

You can also see that the rotation of the original sprite θ_I2 is 45°.

So the angle between the original sprite and the line is θ_I2 − θ_L, which equals 10°.

Once again, the angle between each sprite and the line needs to be equal and opposite. That is:

θ_I2 − θ_L = −1 * (θ_R2 − θ_L)

To find the rotation needed for the reflected sprite, just rearrange that equation to this:

θ_R2 = 2θ_L − θ_I2

Putting in the values for θ_L and θ_I2 gives:

θ_R2 = 2*35° − 45° = 25°

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So in this example the rotation of the reflected sprite θ_R2 is calculated to be 25°, which you can see is correct from the image. If you want to double check though, notice that the difference in angle between the reflected sprite and the line (θ_R2 − θ_L) is −10°, which is equal and opposite to θ_I2 − θ_L. Just remember to flip the reflected sprite before you apply this rotation (assuming the green arrow is the axis you flip it around).